The horizontal lift of the Riemannian Hessian of \(f\)  under the metric \(\bar{g}\)  (REF ) is given by [1]}
\(\overline{\operatorname{Hess}f\left(\left[\bar{{\cal X}}\right]\right)\left[\xi _{\left[\bar{{\cal X}}\right]}\right]} = {\cal P}_{\bar{{\cal X}}}^{{\cal H}}\left(\operatorname{Hess}\bar{f}\left(\bar{{\cal X}}\right)\left[\bar{\xi }_{\bar{{\cal X}}}\right]\right),\) 
